Two squares are chosen at random on a chessboard. What is the probability that they have a side in common?
Two squares are chosen at random on a chessboard. What is the probability that they have a side in common? I have got the total no of events by using 64 C 2. But I am unable to find the numerator(no. of favorable events).
We may count the couples of adjacent squares in the following way: for any square, we may consider how many adjacent squares there are, sum everything, divide by two: $$ \frac{4\cdot 2+ 24\cdot 3+36\cdot 4}{2}=112. $$ The couples of squares are $\binom{64}{2}=2016$, hence the wanted probability is $$ \frac{112}{2016}=\color{red}{\frac{1}{18}}.$$